They don't
The people from Beyond-The-Wall call themselves "Free Folk", almost exclusively.
Evidence
There are only two times that a "wildling" uses the word "wildling" to refer to themselves (within the books), and both are in A Dance With Dragons#.
The first one is in the scene where "Rattleshirt" (actually Mance "glamoured" as Rattleshirt, so we'll call him "RattleMance") is talking to Jon and Mel, and he says:
“Are we talking about betrayals? What was the name of that wildling wife of yours, Snow? Ygritte, wasn’t it?” The wildling turned to Melisandre.
-A Song of Ice and Fire: Book Five - A Dance With Dragons, Chapter Thirty-One (Melisandre I).
This could easily be a slip of the tongue by the author, or even a turn-of-phrase by RattleMance, since he's in the company of "Southerners"@ and he's talking to a "Southerner".
The other is Tormund speaking to Jon after Jon gave his speech about going after Ramsay at the hall:
"Well spoken, crow. Now bring out the mead! Make them yours and get them drunk, that's how it's done. We'll make a wildling o' you yet, boy. Har!"
-A Song of Ice and Fire: Book Five - A Dance With Dragons, Chapter Sixty-Nine (Jon XIII).
Again, in this case, Tormund is in the "South" and he's talking to a "Southerner"; using their venacular.
The University of Ygritte
Additionally, Jon Snow explains to his sort-of cousin, Alys Karstark, that they call themselves "Free Folk":
"So," said Alys, as Jon poured, "I am now a woman wed. A wildling husband with his own little wildling army."
"Free folk is what they call themselves. Most, at least. The Thenns are a people apart, though. Very old." Ygritte had told him that. You know nothing, Jon Snow.
-A Song of Ice and Fire: Book Five - A Dance With Dragons, Chapter Fourty-Nine (Jon X).
See, he does know some things!
Show-canon
This brings us to the show - Game of Thrones - within which the Free Folk seem to constantly call themselves "wildlings".
I chalk this up to two reasons:
- The show writers/creators don't want to confuse people. And yes I will withhold my rant about how they think viewers are 'dumb'.
- The show writers/creators don't see or understand the nuances in the cultures.
Both are equally as likely.
# I know this because I searched my e-books for the term "wildling" and read each part to see who was saying the word. I also use A Search of Ice and Fire.
@ To the Free Folk, anything below the Wall is considered "South", as explained by Ygritte in A Clash of Kings:
“Bael the Bard made it,” said Ygritte. “He was King-beyond-the--Wall a long time back. All the free folk know his songs, but might be you don’t sing them in the south.”
“Winterfell’s not in the south,” Jon objected.
“Yes it is. Everything below the Wall’s south to us.”
He had never thought of it that way. “I suppose it’s all in where you’re standing.”
“Aye,” Ygritte agreed. “It always is.”
-A Song of Ice and Fire, Book Two: A Clash of Kings, Chapter Fifty-One (Jon VI).